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Engineering discrete local dynamics in globally driven dual-species atom arrays
Francesco Cesa, Andrea Di Fini, David Aram Korbany, Roberto Tricarico, Hannes Bernien, Hannes Pichler, Lorenzo Piroli
TL;DR
Globally driven neutral-atom arrays lack straightforward local interaction control, limiting their use for discrete local dynamics. This work uses static dual-species layouts, species-selective driving, and mediated gates to engineer QCA models, including kicked-Ising, Floquet Kitaev, and Hamiltonian digitization. The resulting framework supports chaos detection with minimal demonstrated experimental tools and distinguishes chaotic from non-ergodic dynamics in representative regimes.
Problem
Global lasers address neutral-atom arrays uniformly, while engineering discrete local dynamics requires local interaction control.
Method
The method places data atoms on model vertices and ancilla gadgets on bonds, then alternates species-selective global pulses to mediate local gates in static arrays.
Results
The construction realizes QCA examples including kicked-Ising, Floquet Kitaev, and arbitrary local-Hamiltonian digitization, and its chaos indicator discriminates chaotic from non-ergodic dynamics.
Takeaways & Limitations
Globally driven dual-species arrays can study effectively discretized, circuit-like many-body dynamics beyond native Hamiltonian evolution with minimal experimental control requirements.
Abstract
from arXiv · showhide
We introduce a method for engineering discrete local dynamics in globally-driven dual-species neutral atom experiments, allowing us to study emergent digital models through uniform analog controls. Leveraging the new opportunities offered by dual-species systems, such as species-alternated driving, our construction exploits simple Floquet protocols on static atom arrangements, and benefits of generalized blockade regimes (different inter- and intra-species interactions). We focus on discrete dynamical models that are special examples of Quantum Cellular Automata (QCA), and explicitly consider a number of relevant examples, including the kicked-Ising model, the Floquet Kitaev honeycomb model, and the digitization of generic translation-invariant nearest-neighbor Hamiltonians (e.g., for Trotterized evolution). As an application, we study chaotic features of discretized many-body dynamics that can be detected by leveraging only demonstrated capabilities of globally-driven experiments, and benchmark their ability to discriminate chaotic evolution.
I. INTRODUCTION
The paper engineers discrete local dynamics and QCA in dual-species atom arrays using global, species-selective driving and static layouts. Mediated gates, alternating data and ancilla pulses, and blockade physics convert uniform analog controls into effectively local circuit dynamics.
- I. INTRODUCTION: Discrete local dynamics evolve through strictly local update rules in discrete time and can reproduce complex interacting many-body behavior.The considered QCA use a constant number of translation-invariant layers of single- and nearest-neighbor two-qubit gates.
- I. INTRODUCTION: Global lasers ordinarily constrain neutral-atom experiments to analog evolution, motivating static-layout protocols that provide effective local control without rearrangement.Optical-tweezer arrays offer arbitrary geometries and native Rydberg interactions, but excitation is often driven uniformly.
- I. INTRODUCTION: The construction uses dual-species arrays, species-selective global pulses, and ancillas placed on bonds to mediate translation-invariant discrete dynamics with constant space and time overhead.Data atoms occupy model vertices, while ancillary gadgets occupy bonds and are activated by alternating global pulses.
- I. INTRODUCTION: The framework targets kicked-Ising, Floquet Kitaev honeycomb, and generic local-Hamiltonian digitization models, while enabling chaos indicators accessible with globally-driven experiments.The chaos indicator is reported to discriminate chaotic from non-ergodic dynamics using demonstrated dual-species capabilities.
- I. INTRODUCTION: Alternating data and ancilla driving implements single-qubit operations and mediated entangling gates while returning ancillas deterministically to |g⟩.The resulting effective circuit acts on the data atoms only, with entangling gates mediated by ancillas.
- I. INTRODUCTION: Rydberg blockade dynamics realize the PXP regime, in which an atom evolves only when neighboring atoms are in |g⟩, while dual-species interactions can suppress blockade violations.The authors report that violations can be strongly suppressed and are substantially smaller than in single-species implementations.
B. Dual-species gadgets for mediated gates
Mediated gates use globally driven ancillas to imprint conditional phases on neighboring data atoms. Decorated gadgets extend this mechanism by using blockade-confined superatoms whose size controls the available gate phases.
- B. Dual-species gadgets for mediated gates: A mediator ancilla driven around a closed Bloch-sphere trajectory returns to |g⟩ while acquiring a phase conditional on both neighboring data atoms being in |g⟩.The conditional geometric phase becomes an entangling phase on the data atoms.
- B. Dual-species gadgets for mediated gates: The resulting mediated gate is maximally entangling when ϕ = π.The phase is reabsorbed into the data wavefunction after the ancilla returns to its initial state.
- B. Dual-species gadgets for mediated gates: Driving many elementary gadgets in parallel applies CZ(ϕ) to all neighboring data pairs while leaving the ancillas in their ground-state product state.Each ancilla acquires the phase only when both adjacent data atoms are in |g⟩.
- B. Dual-species gadgets for mediated gates: Superatom size controls the phase of the mediated CZ gate, allowing different bonds to implement different CZ(ϕSi,j) gates.For the constructions considered, the required maximum gadget size is Smax ≤3.
- B. Dual-species gadgets for mediated gates: A blockade-confined cluster of S ancillas behaves as a two-level superatom with enhanced Rabi frequency SΩ.The collective states are |GS⟩ and a single-excitation W-state |RS⟩.
- B. Dual-species gadgets for mediated gates: Appropriate ancilla driving can simultaneously close trajectories for multiple superatom sizes and imprint arbitrary target phases in parallel.The control schedule is implemented through the phase profile ξ(t).
3. Parallel quantum control of decorated gadgets
Decorated gadgets are controllable enough to implement synchronous QCA and representative discrete many-body models with globally applied pulses. Their parallelizability determines circuit depth, while gadget sizes encode interaction inhomogeneity.
- 3. Parallel quantum control of decorated gadgets: Complete controllability guarantees a finite control phase ξ(t) that connects any two product states of superatoms with different cardinalities.This establishes the existence of schedules that realize prescribed phases across multiple gadget sizes.
- 3. Parallel quantum control of decorated gadgets: Time-optimal controls are used to reduce errors from the finite lifetime of Rydberg states, with Smax = 3 sufficient for the Floquet Kitaev construction.GRAPE is used to compute the time-optimal controls.
- A. Synchronous dynamics: Synchronous QCA require only two pulses: a data pulse updating local frames and an ancilla pulse activating the commuting gadgets.The kicked-Ising model is an example of this synchronous class.
- 1. The kicked-Ising model: For the kicked-Ising model, data rotations implement the kick and ancilla gadget pulses implement the Ising layer through mediated gates.The construction applies a detuned data pulse followed by ancillary pulses realizing CZ(ϕ) on neighboring atoms.
2. Inhomogeneous kicked-Ising model
The construction extends from homogeneous synchronous dynamics to inhomogeneous and asynchronous QCA. Decorated gadgets encode direction-dependent couplings, while orientation-specific gadget sizes implement the Floquet Kitaev honeycomb steps.
- 2. Inhomogeneous kicked-Ising model: An inhomogeneous kicked-Ising model assigns different couplings Jx and Jy to the two lattice directions, thereby breaking isotropy.The x and y bonds are implemented with elementary gadgets and size-S = 2 superatoms, respectively.
- 2. Inhomogeneous kicked-Ising model: The data pulse uses α′ = hτ + 2τ(Jx + Jy), while ancillary phases ϕx = −4τJx and ϕy = −4τJy implement the directional CZ gates.The phase assignments distinguish x- and y-oriented nearest-neighbor bonds.
- B. Asynchronous QCA: Asynchronous QCA have noncommuting local updates and therefore cannot apply all interacting updates in parallel.The Floquet Kitaev honeycomb model and digitized arbitrary 2-local Hamiltonians are presented as examples.
- 1. The Floquet Kitaev honeycomb: Because the Floquet terms do not commute, the effective Floquet Hamiltonian approximates the Kitaev honeycomb Hamiltonian only for small τ.At finite τ, O(τ) terms can break time-reversal symmetry and open a gap in the otherwise gapless non-Abelian phase.
- 1. The Floquet Kitaev honeycomb: The Floquet Kitaev implementation uses gadget sizes SZ = 1, SY = 2, and SX = 3 for the three bond orientations.Each step combines basis changes on the data with an orientation-selective mediated-gate layer.
2. Digitization of arbitrary local Hamiltonians
The construction digitizes translation-invariant nearest-neighbor Hamiltonians using species-alternated global pulses and bond gadgets. It supports standard Trotterized simulation at small time steps while also realizing arbitrary-step discrete dynamics, with implementation costs tied separately to parallelizability and homogeneity.
- The target Hamiltonian is locally connected, symmetric, and identical on every bond, enabling a translation-invariant nearest-neighbor digitization.
- For small τ, the digitization has error ε = O(Nτ 2), enabling standard Trotterized simulation.
- The same implementation realizes UH for arbitrary τ, extending beyond Trotterization to more generic discrete dynamics derived from H.
- Each α step rotates data qubits, applies the bond interaction through activated ancilla gadgets, and reverses the rotation.
- One application of UH requires 6 species-alternated laser pulses after merging adjacent sub-steps.
- Circuit depth depends on whether updates parallelize, whereas superatom-size overhead depends on spatial homogeneity.
V. OBSERVING QUANTUM CHAOS
The paper develops a coarse-grained operator-spreading diagnostic for quantum chaos that requires fewer experimental resources than OTOCs. The quantity gO(t) approaches a chaos-specific stationary limit and exhibits finite-time decay governed by light-cone growth, while exponential decay alone is not conclusive.
- Quantum chaos involves phenomena including information scrambling, eigenstate thermalization, and random-matrix behavior, but is experimentally difficult to diagnose through quantities such as OTOCs.
- The diagnostic gO(t) extracts coarse-grained information about operator dynamics and can discriminate dynamical regimes using experimentally demonstrated dual-species capabilities.
- gO(t) is built from the operator-size distribution of O(t), whose Pauli-string coefficients encode how the operator spreads across the system.
- For chaotic dynamics, gO(t) approaches a stationary limit when the operator support reaches the full system, A(t) ≃ N.
- At finite times, chaotic dynamics produce exponential decay of gO(t) as the light cone grows with A(t) ∼ (vt)^D.
- Finite-time exponential decay alone cannot establish chaos, whereas deviations from the predicted form can signal its absence, including polynomial decay from logarithmic light-cone growth.
B. Measuring signatures of quantum chaos in neutral atom experiments
The measurement protocol replaces extensively disordered Haar-random product states with a four-state single-qubit 2-design and prepares them using global pulses plus patterned tweezers. This makes unbiased estimation of gO(t) compatible with demonstrated globally driven neutral-atom capabilities, subject to observable and control limitations.
- Extensive independent Haar randomness is inefficient to prepare with global controls, motivating a reduced-disorder state ensemble.
- A single-qubit 2-design reproduces the Haar second moment using four tetrahedral states on the Bloch sphere.
- Global unitaries and progressively deactivated tweezers prepare the assigned states by freezing addressed atoms while the others evolve.
- The protocol randomly assigns tetrahedral states, estimates ⟨ψ|O(t)|ψ⟩2 through repeated measurements, and averages over M product-state choices to estimate gO(t).
- Sampling product states from this four-state ensemble reduces the initial disorder from extensive to a small local choice while preserving the required second moment.
- The protocol is efficient with global controls only when O is uniform across its support, and current tweezers cannot provide mid-circuit local control without decoherence and related errors.
C. Observing quantum chaos: numerical analysis
The generating function gO(t) distinguishes integrable and chaotic behavior in the 1D kicked Ising model and captures different operator-spreading regimes in the Floquet Kitaev model. However, early-time decay does not always separate chaotic from non-ergodic dynamics, and two-dimensional simulations remain limited to shallow times.
- 1D kicked Ising: gO(t) distinguishes the chaotic and free-fermionic kicked-Ising regimes, with the integrable case showing larger magnitudes and strong revivals.The simulations use 1000 random initial configurations for N = 8 and N = 16 sites; the N = 8 operator-size distribution also confirms the generating-function relation.
- Floquet Kitaev honeycomb: In the Floquet Kitaev model, gO(t) detects the slower operator spreading of a quasi-1D regime compared with genuinely two-dimensional dynamics.This distinction is visible even during the early-time window accessible in the simulations.
- Floquet Kitaev honeycomb: For the Kitaev Floquet QCA, gO(t) captures dynamical features and distinguishes regimes using relatively shallow experimental protocols.The reported analysis corresponds to approximately three applications of the Floquet update.
- Limitations: Early-time exponential decay can also occur in a non-ergodic Clifford-circuit regime, so only late-time behavior unambiguously distinguishes chaos there.The early decay is followed by revivals, while the late-time regime remains necessary for an unambiguous diagnosis.
- Limitations: In two dimensions, simulations did not extend beyond small times, leaving a more thorough study of quantum chaos to future experimental implementation.The limitation applies particularly to the late-time dynamics of the two-dimensional QCA.
Appendix A: Violations from the ideal blockade regime
The appendix estimates unwanted interspecies interactions in the blockade construction and explains controllability of superatoms with distinct cardinalities. The idealized control model is justified when these interactions remain sufficiently suppressed.
- Interaction estimates: The strongest unwanted interaction pairs each data atom with the nearest non-neighbor atom of the opposite species.The estimate focuses on the honeycomb geometry and assumes interspecies interactions dominate intraspecies interactions.
- Interaction estimates: Assuming a van der Waals tail V_i,j ∼ d^-6, the interaction ratio can be estimated from the relevant atom separations.
- Interaction estimates: The unwanted-to-blockade interaction ratio is smaller than 0.0156, the value reported for established single-species 1D PXP implementations.This comparison is used to justify the Rydberg-blockade approximation and the extension to other geometries.
- Interaction estimates: The estimate assumes dominant interspecies interactions and a van der Waals potential, although the full interspecies potential has a crossover between 1/R^3 and 1/R^6 regimes.Choosing unwanted-interaction distances larger than the crossover distance makes the van der Waals approximation more appropriate.
- Complete controllability: For superatoms of distinct cardinalities, the available global controls generate independent rotations and yield complete controllability of the product system.The proof uses the Lie algebra generated by the control directions and an invertible Vandermonde matrix when all cardinalities differ.
- Complete controllability: The controllability construction isolates each superatom's generators by solving a linear system M · β(S) = eS, whose invertibility requires distinct superatom cardinalities.
Appendix C: Quantum optimal control
Appendix C develops numerical optimal-control procedures for driving superatoms of different sizes to target phases. It uses GRAPE-based optimization and identifies the shortest duration meeting a prescribed error threshold.
- Control problem: Different superatom sizes have different enhanced Rabi frequencies, making a common phase control ξ(t) non-trivial to design.Time-optimal solutions are preferred because finite Rydberg-state lifetimes increase errors with process duration.
- GRAPE optimization: GRAPE searches for a piecewise-constant global phase that minimizes an error function encoding the target evolution.The control interval is discretized into M segments with phase variables ξ = (ξ1, …, ξM).
- GRAPE optimization: The optimization can use an analytic gradient, including derivatives of the error with respect to the discretized control phases.
- Time optimization: The quantum speed limit Tmin is defined as the smallest duration for which Errmin < 1.01 × 10^-10.The error threshold is Eth = 10^-10, and the minimum-error profile typically drops sharply near Tmin.
- Time optimization: For target phases parameterized by ϕ, Tmin(ϕ) is continuous but can be non-differentiable where distinct local-minimum branches meet.The optimization therefore requires more careful treatment near such points, including multiple starting directions.
Appendix D: Details on the numerical simulation of the QCA
Appendix D describes tensor-network and sampling procedures for simulating the QCA observables and estimating their generating function. The simulations are controlled but restricted to small systems or early times by growing tensor-network costs.
- Simulation methods: The numerical results use tensor-network state-vector simulations implemented with ITensor, using MPS for one-dimensional and PEPS for two-dimensional dynamics.
- Simulation methods: The simulations are numerically exact within significant digits in the reported regimes, but this restricts calculations to small systems or few time steps.
- Sampling and uncertainty: The generating function is estimated from sampled observables, with uncertainty obtained from ten independent batches corresponding to repeated experimental runs.The uncertainty region is the standard deviation across the batch estimates.
- One-dimensional simulations: For the kicked-Ising simulations, the exact MPS bond dimension grows as χ_t ≤ min(4^t, χ_max), with χ_max = 2^(N/2).
- One-dimensional simulations: For chaotic dynamics, the late-time generating-function value approaches 1/(2^N + 1), requiring numerical errors much smaller than 2^-N to resolve it.For N = 8, MPS results show perfect agreement with exact calculations.
- Observable distributions: The sampled observable distribution becomes approximately Gaussian after a few chaotic time steps, whereas free-fermion dynamics show non-Gaussian deviations near recurrences.
- Two-dimensional simulations: Two-dimensional PEPS simulations are limited because bond dimensions grow exponentially with time and expectation values require approximate boundary-MPS contractions.The expectation-value cost can scale as O(χ^14), severely restricting the number of simulated time steps.
- Two-dimensional simulations: Restricting evolution to the observable's light cone reduces cost, although the Kitaev Floquet light cone grows to 8, 28, and 60 qubits at t = 1, 2, and 3.Intermediate sub-steps are also considered to track the three component evolutions of the Floquet operator.
Appendix E: Higher-order moments of the tetrahedral ensemble
Appendix E derives higher-order moments for the tetrahedral ensemble by expanding tensor products into Pauli components and organizing terms by site subsets. The resulting moment operator is expressed through tetrahedral moment tensors and Pauli-operator products.
- Moment construction: The appendix computes a general expression for the k-th moment of the tetrahedron ensemble.
- Moment construction: The tetrahedral ensemble uses Bloch-sphere vectors pointing to the vertices of a regular tetrahedron.
- Moment construction: Tensor products are expanded by selecting subsets of sites carrying Pauli operators while the identity acts on the remaining sites.The expansion is ordered by the subset cardinality |E| and then by Pauli components.
- Moment tensors: The moment tensors are defined from products of the tetrahedral Bloch vectors and provide the building blocks for higher-order ensemble moments.
- Moment decomposition: Combining the expansion identities yields a compact decomposition of N_k into tetrahedral moment tensors and tensor products of Pauli operators.
1. Explicit moments up to fourth order
The paper explicitly evaluates moment operators N_k through fourth order using a tetrahedral choice of directions. The resulting moments decompose into tensor-order contributions, with lower-order cases simplifying by symmetry.
- Overview: Moment operators N_k are explicitly evaluated through fourth order by direct application of Eq. (E4).The construction gives explicit expressions for k ≤ 4.
- First order: For first order, symmetry makes all first-order moment tensors vanish, yielding N_1 = I.
- Second order: At second order, the only nonvanishing moment tensor is diagonal.
- Third order: At third order, N_3 contains tensor-order contributions m = 0, 2, and 3 and is decomposed using permutations from the symmetric group of order 3.The permutation group acts on {x, y, z}.
- Fourth order: At fourth order, N_4 contains tensor-order contributions m = 0, 2, 3, and 4 and is decomposed using distinct permutations of multisets such as {a, a, b, b}.Perm(a, a, b, b) denotes all distinct permutations of that multiset.